Question:
If $f(x)=x+1$, and $g(x)=x^{3}$, then $f^{-1}(g(x))$ is equal to
If $f(x)=x+1$, and $g(x)=x^{3}$, then $f^{-1}(g(x))$ is equal to
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Logic Tip: Function composition is evaluated from the inside out. Always identify the inner function ($g(x)$) and the outer operation ($f^{-1}$) independently before combining them to prevent algebraic errors.
Updated On: Apr 27, 2026
- $x-1$
- $x^{3}+2$
- $x+1$
- $x^{3}+x$
- $x^{3}-1$
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The Correct Option is
Solution and Explanation
Concept:
To evaluate $f^{-1}(g(x))$, we must first determine the inverse of the function $f(x)$, denoted as $f^{-1}(x)$, and then substitute $g(x)$ into it.
Step 1: Find the inverse function of f(x).
Let $y = f(x)$. $$y = x + 1$$ Swap $x$ and $y$ to solve for the inverse: $$x = y + 1$$ $$y = x - 1$$ Therefore, the inverse function is: $$f^{-1}(x) = x - 1$$
Step 2: Substitute g(x) into the inverse function.
We are tasked with finding $f^{-1}(g(x))$. We take our result from Step 1 and replace $x$ with $g(x)$: $$f^{-1}(g(x)) = g(x) - 1$$
Step 3: Insert the expression for g(x).
Given $g(x) = x^3$, substitute this into the equation from Step 2: $$f^{-1}(g(x)) = x^3 - 1$$
To evaluate $f^{-1}(g(x))$, we must first determine the inverse of the function $f(x)$, denoted as $f^{-1}(x)$, and then substitute $g(x)$ into it.
Step 1: Find the inverse function of f(x).
Let $y = f(x)$. $$y = x + 1$$ Swap $x$ and $y$ to solve for the inverse: $$x = y + 1$$ $$y = x - 1$$ Therefore, the inverse function is: $$f^{-1}(x) = x - 1$$
Step 2: Substitute g(x) into the inverse function.
We are tasked with finding $f^{-1}(g(x))$. We take our result from Step 1 and replace $x$ with $g(x)$: $$f^{-1}(g(x)) = g(x) - 1$$
Step 3: Insert the expression for g(x).
Given $g(x) = x^3$, substitute this into the equation from Step 2: $$f^{-1}(g(x)) = x^3 - 1$$
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